J. Fluid Mech. (2008), vol. 596, pp. 133–142. c© 2008 Cambridge University Press

doi:10.1017/S0022112007009536 Printed in the United Kingdom

133

Transition to turbulence in duct flow

DAMIEN BIAU, HOUSSAM SOUEID†AND ALESSANDRO BOTTARO

Universita di Genova, DICAT, Via Montallegro 1, 16145 Genova, Italy

(Received 11 July 2007 and in revised form 7 November 2007)

The transition of the flow in a duct of square cross-section is studied. Like inthe similar case of the pipe flow, the motion is linearly stable for all Reynoldsnumbers; this flow is thus a good candidate to investigate the ‘bypass’ path toturbulence. Initially the so-called ‘linear optimal perturbation problem’ is formulatedand solved, yielding optimal disturbances in the form of longitudinal vortices. Suchoptimals, however, fail to elicit a significant response from the system in the nonlinearregime. Thus, streamwise-inhomogeneous sub-optimal disturbances are focused upon;nonlinear quadratic interactions are immediately caused by such initial perturbationsand an unstable streamwise-homogeneous large-amplitude mode rapidly emerges. Thesubsequent evolution of the flow, at a value of the Reynolds number at the boundarybetween fully developed turbulence and relaminarization, shows the alternance ofpatterns with two pairs of large-scale vortices near opposing parallel walls. Such edgestates bear a resemblance to optimal disturbances.

1. IntroductionTransition to turbulence in ducts is still an unsolved issue despite the more than

120 years since the observations by Osborne Reynolds that led to the definition ofa similarity parameter, the ratio of the viscous to the convective time scale, capableof broadly separating the cases where the flow state was laminar from those whereturbulence prevailed. Recent years have seen a resurgence of interest in the topic,spurred by new developments in linear and nonlinear stability theories. As is nowwell known, classical small-perturbation theory is not able to provide an explanationfor the onset of transition in ducts and pipes (Gill 1965; Salwen, Cotton & Grosch1980; Tatsumi & Yoshimura 1990). Current understanding ascribes the failure ofclassical theory to its focus on the asymptotic behaviour of individual modes; when asmall disturbance composed of a weighted combination of linear eigenfunctionsis considered, there is the potential for very large short-time amplification ofperturbation energy, even in nominally stable flow conditions. This behaviour hasbeen reported by Landahl (1980) and Boberg & Brosa (1988) and has been giventhe name of algebraic instability (and later ‘transient growth theory’), since the initialrapid growth in time of small disturbances is proportional to t . The property is relatedto the non-normality of the linearized stability operator (which does not commutewith its adjoint).

Despite the appeal and elegance of transient growth theory, it was realized thatthe fully nonlinear Navier–Stokes equations need to be used to understand transition

† Also at the Institut de Mecanique des Fluides de Toulouse, Allee du Pr. C. Soula, 31400Toulouse, France.

134 D. Biau, H. Soueid and A. Bottaro

phenomena. In this context, we mention Nagata (1990, 1997), Waleffe (1997, 1998,2003), Faisst & Eckhardt (2003) and Wedin & Kerswell (2004), who present travellingwave and equilibrium solutions of the Navier–Stokes equation for channel and pipeflows that are possibly related to transition. Experimental investigations along theselines are due to Hof et al. (2004, 2005).

In dynamical systems’ terminology it is argued that unstable travelling wavesappear through saddle node bifurcations in phase space; the travelling waves foundso far are all saddle points with low-dimensional unstable manifolds. The saddlesact by attracting the flow from the vicinity of the laminar state, and then repelling.For transitional or turbulent, yet moderate, values of the Reynolds number Re theflow wanders in phase space between a few repelling states, spending much time intheir vicinities, before being abruptly ejected, so that experimental observations yieldrecurrent sequences of familiar patterns (Artuso, Aurell & Cvitanovic 1990; Kerswell2005).

A yet unresolved issue concerns the initial conditions that are most suited to yieldsuch unstable states. Traditional emphasis on so-called optimal perturbations maybe misplaced. In fact, there is but a weak connection between the flow structuresthat grow most during the linear transient phase and the chaotic flows found atlarge times. Such a connection for the case of pipe flow concerns the so-called edgestate which sits on a separatrix between laminar and turbulent flows (Eckhardt et al.2007; Pringle & Kerswell 2007). This state, made up of two asymmetric vorticesin the cross-section, resembles the optimal disturbance of transient growth theory(Bergstrom 1993). For the motion in a square duct there seems to be no connectionat all: the low-Re turbulent flow, when averaged in time and space, is characterizedby eight secondary vortices symmetric about diagonals and bisection lines. It seemsreasonable to argue that such secondary structures represent the skeleton of theunstable periodic orbits, but the disturbances that grow most in the linear transientphase are formed by two vortices, symmetric about a diagonal (Galletti & Bottaro2004). In both configurations, pipe and square duct, the optimal perturbation is astationary pseudo-mode, elongated in the streamwise direction, and not a travellingwave. This is a generic occurrence in wall-bounded shear flows, and it does notbode well for the establishment of a simple, direct relation between small-amplitudedisturbances (excited in an initial receptivity phase) and finite-amplitude wave-likestates. Although nonlinear effects can be pinpointed as the missing link betweenearly stage of transition and late stages, there is scope for a receptivity analysisfocused on transiently growing initial conditions, followed by nonlinear simulations.A motivation for the search for wave-like structures during the early stages oftransition is also provided by recent careful experiments (Peixinho & Mullin 2006) onthe reverse transition in pipe flow, where modulated wavetrains are found to emergefrom long-term transients.

The present paper starts by comparing the efficiency of optimal and sub-optimalperturbations in triggering transition to turbulence at a value of Re at the thresholdbetween laminar and turbulent flow; it then shows that the turbulent motion oscillatesaround edge states which display an intriguing resemblance to optimal disturbances,before relaminarization occurs. Finally, an interpretation of the results is providedafter projecting them onto a suitably defined phase space.

2. Model configurationThe incompressible flow in a duct of square cross-section is an appealing

configuration for the presence of geometrical symmetries capable of strongly

Transition to turbulence in duct flow 135

constraining the patterns of motion. Countless studies have been devoted to theformation of secondary vortices in the turbulent regime (see Gavrilakis 1992 fora direct numerical simulation approach) and, more recently, an attempt has beenmade to link the appearance of such large-scale coherent states to the vorticesappearing during the initial optimal transient phase of disturbance growth (Galletti& Bottaro 2004; Bottaro, Soueid & Galletti 2006). The longitudinal laminar flowvelocity component has an analytic form U (y, z) available, for example, in Tatsumi& Yoshimura (1990), with y and z cross-stream axes. After normalizing distanceswith the channel height h, velocities with the friction velocity uτ , with u2

τ = (−h/4ρ)(dP/dx), time with h/uτ and pressure with ρu2

τ , the following equations are found togovern the behaviour of the developed flow in an infinite duct:

ux + vy + wz = 0,

ut + uux + vuy + wuz = −px + �u/Reτ + 4,

vt + uvx + vvy + wvz = −py + �v/Reτ ,

wt + uwx + vwy + wwz = −pz + �w/Reτ ,

with �= ∂xx + ∂yy + ∂zz and Reτ = uτ h/ν. By using uτ as velocity scale we fix thepressure gradient, rather than the flow rate.

An incompressible pseudo-spectral solver, based on Chebyshev collocation in y

and z and Fourier transform along x, has been employed to solve these equations.For time-integration a third-order semi-implicit backward differentiation/Adams–Bashforth scheme is used. In order to compute a pressure unpolluted by spuriousmodes, the pressure is approximated by polynomials (PN−2) of two units lower-order than for the velocity (PN ). Only one collocation grid is used, and no pressureboundary condition is needed. The accuracy and stability properties of the methodare discussed by Botella 1997. An adequate grid at Reτ = 150 has been found to becomposed of 51 × 51 Chebyshev points, with Nx = 128 streamwise grid points or 84Fourier modes after de-aliasing. The x-length of the domain has been chosen equalto 4π to accommodate a sufficiently large range of wavenumbers α, with periodicboundary conditions. A finer grid resolution has also been used for fully developedturbulent flow with 71 × 71 × 256 physical grid points, or 71 × 71 × 170 spectralmodes and a streamwise length Lx =6π (with this resolution we obtain an excellentmatch with the results by Gavrilakis 1992 for Reτ = 300). The finer resolution runprovides a slightly larger value of the threshold energy for transition, but integralquantities such as disturbance energy or skin friction factor are only marginallyaffected. Since the threshold value is, in any case, a function also of the shape ofthe initial condition, we do not deem it necessary to pursue expensive calculations todetermine it exactly. For all cases studied, an adequate time step has been found to be�t = 5 × 10−4.

The mean value of the generic function g is defined as g(y, z) = (1/Lx T )/∫xt

g(x, y, z, t) dx dt . The addition of time averaging is necessary because of the

finite (relatively low) streamwise length. The bulk velocity is Ub =∫

yzu dy dz and the

centreline velocity is Uc = u(0.5, 0.5). The friction factor for the square duct can bewritten as f = 8 u2

τ /U 2b . Some representative results are given in table 1, and the

secondary flow field at Reτ =150, averaged over forty units of time, is shown infigure 1. It displays a very regular pattern with eight vortices, despite the fact that noaveraging over quadrants has been performed.

136 D. Biau, H. Soueid and A. Bottaro

f Uc/Ub Reb Rec

Laminar 0.018 2.0963 3163 6630.4Turbulent 0.0415 1.53 2084 3188

Table 1. Comparison of some numerical values for laminar and fully developed turbulent flowat Reτ = 150. The subscript b refers to bulk and c to centreline. The skin friction f given bythe empirical correlation by Jones (1976) is f = 0.0481.

0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

z

y

Figure 1. Turbulent mean crossflow vortices and streamwise flow contours; isolines arespaced by 4uτ .

3. Optimal perturbationsThe Navier–Stokes equations, linearized around the ideal laminar flow, are

iαu + vy + wz = 0,

ut + iαUu + vUy + wUz = −iαp + (−α2u + uyy + uzz)/Reτ ,

vt + iαUv = −py + (−α2v + vyy + vzz)/Reτ ,

wt + iαUw = −pz + (−α2w + wyy + wzz)/Reτ ,

associated with boundary conditions u = v = w =0 on the walls; α is the streamwisewavenumber. The equations are integrated from a given initial condition at t = 0 upto a final target time t = T . To identify the flow state at t = 0 producing the largestdisturbance growth at any given T , a variational technique, based on the repeatednumerical integration of direct and adjoint stability equations, is used, coupled withtransfer and optimality conditions (Corbett & Bottaro 2000). The functional for whichoptimization is sought is based on an energy-like norm and is

G(T ) =E(T )

E(0), with E =

1

2

∫y

∫z

(u∗u + v∗v + w∗w) dy dz,

with ∗ denoting complex conjugate.As a preliminary result the energy stability limit for this flow is determined; the

minimal Reynolds number below which disturbances decrease monotonically is foundto be Reτ = 23.22 for α = 0. In terms of Reynolds number based on centreline velocityand half the channel height, this limiting value is Rec = 79.44. For comparison, in the

Transition to turbulence in duct flow 137

1 2 30

200

400

600

800

1000

t

G

α = 0α = 0α = 1α = 2

Figure 2. Crossflow velocity vectors for the optimal disturbances for α = 0, 1 and 2. Notethat for α �= 0 the streamwise velocity fluctuations do not vanish at t = 0.

case of Poiseuille flow, Rec � 49, with α =0 and spanwise wavenumber β � 2 (cf.Schmid & Henningson 2001).

Then, we compute optimal perturbations at Reτ = 150, for three different Fouriermodes: α = 0, 1, 2. The gain and the corresponding crossflow optimal disturbancesat t = 0 are presented in figure 2. A global optimal solution (G = 873.11) is foundfor α = 0 at a time T = 1.31 and a comparable gain (G = 869.03) is found at a latertime (T = 2.09) for a solution which is topologically different. These two solutions,which we call ‘global optimals’ represent streamwise vortices of vanishing streamwisedisturbance velocity; they evolve downstream producing streaks of high and lowlongitudinal velocity. A global optimal state with two cells arranged along a ductdiagonal, was obtained by Galletti & Bottaro (2004) in the context of a spatial, ratherthan temporal, optimization strategy. The existence of a four-cell optimal was notreported. It will be shown below that the two- and the four-cell states are very robust:when they are used as initial conditions for nonlinear simulations the flow trajectorycannot evolve away from them towards a different topology which eventually exploitsother symmetries, except when exceedingly large disturbance energies are given asinput.

For non-zero streamwise wavenumber, sub-optimal perturbations are found whichtake the form of modulated travelling waves. For α = 1, G =381.41 at T = 0.384 andfor α = 2, G = 237.05 at T = 0.246. To represent a wavetrain the temporal dependenceof the generic disturbance can be written as the product of an exponential wave partand an envelope function slowly modulated in time: q(x, y, z, t) = q(y, z, t)eiα(x−ct).The phase velocity is found to be quasi-constant with time and close to thebulk velocity: c(α = 1) = 1.1117 and c(α = 2) = 1.1536, both scaled with Ub. Thestudy of the temporal (rather than the spatio-temporal) evolution of disturbances isacceptable when flow structures travel at a well-defined speed within the duct. Such anapproximation appears to be reasonably well satisfied in experiments on equilibriumpuffs in pipe flow (Hof et al. 2005), which are found to be advected downstreamat speeds slightly larger than Ub, for values of the Reynolds number Reb = UbD/ν

138 D. Biau, H. Soueid and A. Bottaro

0 5 10 15 200.01

0.02

0.03

0.04

0.05

t

f

α = 0α = 0α = 1α = 2

Figure 3. Skin friction for the four different optimal perturbations. For α = 0, the initialenergy E0 is equal to 10−1, for α = 1, E0 = 7.8 × 10−3 and for α =2, E0 = 4.4 × 10−3.

(D pipe diameter) exceeding Reb ≈ 1800. Below such a threshold, turbulence can nolonger be maintained autonomously.

4. Nonlinear evolutionTemporally evolving simulations have been conducted in a periodic duct of length

4π for a variety of initial conditions at Reτ =150, a value very close to the threshold ofself-sustained turbulence, as confirmed independently by Uhlmann et al. (2007). Foreach direct numerical simulation the initial state consists of the quasi-parabolic baseflow profile plus a optimal or sub-optimal perturbation, normalized with prescribedenergy E0, plus random noise. The complex amplitude of the noise in Fourier spacevaries between ±10−10; the noise is necessary to fill the spectrum.

In figure 3 the time evolution of the skin friction factor f is plotted for four initialconditions of different initial energies. When the simulations are initiated with oneof the two global optimal solutions of figure 2 the ensuing behaviour is uneventful(see figure 3), and even for a rather large initial disturbance amplitude, E0 = 0.1, noinstability appears to modify the flow, which returns slowly to the laminar conditionwith f =0.018. The results are more illuminating when linear travelling waves areused to initiate the nonlinear computations. When the condition at t = 0 is the sub-optimal state with α = 1 or 2, the initial growth of f is slower than for the globaloptimal case but, by t =2, a strong mean flow deviation is created by nonlinearinteractions, leading to a friction factor which oscillates around the turbulent meanvalue f = 0.0415. It is notable that the energies E0 of the sub-optimal initial conditionssufficient to trigger transition are very much lower than 10−1.

We have explored in more detail the long-time behaviour of the flow when thesub-optimal initial perturbation with α = 1 is used to trigger transition; the resultsare summarized in figure 4. As a function of the value of E0, the Navier–Stokescalculations either recover the linear behaviour (when E0 � 5 × 10−3) or departfrom it. The threshold for transition is found for E0 = 7.8 × 10−3: below this initialenergy value the flow returns rapidly (within a few units of time) to the laminarstate; above it the flow becomes turbulent. The energy gain of figure 4 shows thatwhen E0 = 7.8 × 10−3 there is, at t =0.8, a sudden increase of the fluctuation energy,probably linked to an instability of the distorted mean flow. There is an interestingconnection here with the recent theory of ‘minimal defects’ (Bottaro, Corbett &Luchini 2003; Biau & Bottaro 2004; Gavarini, Bottaro & Nieuwstadt 2004; Ben-Dov& Cohen 2007). The secondary flow at t = 0.8 shown in the same figure displays

Transition to turbulence in duct flow 139

0 20 40 60 80t

0 1 2 3 4 5100

101

102

103

104

t

E—E0

Figure 4. Evolution in time of the fluctuation energy, with initial condition given by thesub-optimal perturbation with α = 1. Different curves correspond to different initial energyvalues: solid line E0 = 7.8 × 10−3, dash-dotted line E0 = 7.7 × 10−3; dashed line E0 = 5 × 10−3

and dotted line represents the linear computation. The energy is normalized with its initialvalue E0. The streamwise-averaged vortices are drawn at times t = 0.8, 20, 50, 70 and 80.

symmetries about bisectors and diagonals, but this is not a generic occurrence anddifferent initial conditions generate mean flow defects with other symmetries. ForE0 = 7.7 × 10−3 the growth is followed by decay and rapid relaminarization (see alsofigure 5a); when E0 = 7.8 × 10−3 the growth which starts at t =0.8 is followed by arapid filling of the spectrum with a peak in the intensity of fluctuations at t =1.8.Such a filling is made clear by figure 5(b): first the modes with α = n, n ∈ �, growbecause of quadratic interactions, then the modes with α = (2n − 1)/2, n ∈ �, emergeout of the random noise (visible in the figure after t = 5). Figure 5(a) focuses onthe modes α =0 and α =2 which are the first to be produced by nonlinearities. Thestreamwise-independent mode remains amplified for a long time because of the lift-upeffect, its appearance from the α = ± 1 fundamental mode is the analogue of theso-called oblique transition process in channel flow (Schmid & Henningson 1992).By time t = 3 the turbulent flow can be considered as fully developed; we speculatethat this is an edge state, under the assumption that there is turbulence on the edgesurface in phase space at the smallest possible Re for which it is sustained. Such anedge state persists until t ≈ 78 (see figure 4); it remains dynamically connected to thelaminar base flow solution since relaminarization is abruptly reached at t ≈ 80, aftercoalescence of the smaller vortices into a pair and then into a single large vortex.While turbulence is maintained, the secondary patterns displayed at t = 20, 50 and70 in figure 4 resemble the four-cell global optimal disturbance. At t = 20, two pairsof vortices are clearly visible in the cross-section; they are close to the two verticalwalls, which can thus be defined as ‘active’ since it is there that the turbulent wallcycle operates (Uhlmann et al. 2007). This 4-cells state oscillates while maintainingremarkable coherence for some twenty units of time. Averaging over t yields the sameflow pattern with four regular vortices discovered very recently by Uhlmann et al.(2007). Past t = 50, the active walls shift and the large-scale vortices ‘lean’ on thehorizontal surfaces, despite the presence of smaller intermittent features that can befound near the left vertical wall at t = 50 and the right wall at t = 70. At t ≈ 80 the flowrelaminarizes; tests performed with different lengths of the computational domain

140 D. Biau, H. Soueid and A. Bottaro

0 1 2 310–4

10–2

100

102

(a)104

t

κn

α = 0

α = 1

α = 2

0 2 4 6 810–4

10–3

10–2

10–1

100

101

t

(b)

Figure 5. (a) Temporal behavior of three different Fourier modes (α = 0, 1, 2) for thenonlinear simulation initiated by the optimal α = 1 initial condition with E0 = 7.8 × 10−3

(solid lines) and E0 = 7.7 × 10−3 (dashed lines), plus low-amplitude noise. (b) The evolution ofthe fluctuations (α = 0.5, 1, 1.5, ...) for the case E0 = 7.8×10−3. The power density spectrum κn

is defined by κn = 1/N2∫

yz(u∗u)n +(u∗u)N−n dy dz, with un (αn, y, z, t) the x-Fourier transform

of u, from which the laminar profile has been subtracted.

2000 2400 2800 320050

100

150

200

250

300

350

Reb Reb

EU

α = 0α = 0α = 1α = 2

1900 2000 2100 2200 2300 240090

100

110

120

130

140

150

160

170(a) (b)

Figure 6. ‘Phase diagram’ representation of the transition process: time evolution in Reynoldsnumber, based on bulk velocity, and mean flow energy EU . The simulations start fromthe laminar flow solution (open circle in (a)) plus a (sub-)optimal perturbation. The crosswith arrows is intended to show qualitatively the presence of an unstable saddle node. Thetime-averaged values of EU and Reb in the fully developed turbulent regime are indicated bydashed lines in the zoom (b) which refers only to the initial disturbance perturbations withα = 1.

show that the threshold E0 for transition remains unaffected, but relaminarizationoccurs later both for a shorter computational box (t ≈ 90 when Lx =2π) and a longerbox (t ≈ 150 when Lx = 6π), for the same numerical grid density. Interestingly, forthe case of the short box the oscillations of the variables around the mean displayenhanced amplitudes, highlighting the fact that a distorted dynamics could be causedby constraining the flow structures too much.

For a geometrical description of the transition process, we choose the phasesubspace spanned by two observables: the Reynolds number (based on bulk velocity)and the energy of the streamwise-averaged flow, EU = 0.5

∫yz

U 2 +V 2 +W 2 dy dz, with

(U, V, W ) the velocity vector after streamwise averaging.Figure 6 shows some trajectories between the laminar (Reb =3163, EU = 306.45)

and turbulent (Reb = 2084, EU = 116) fixed points (the latter is defined as the temporal

Transition to turbulence in duct flow 141

average of EU and Reb in the fully developed turbulent regime). The paths for theinitial conditions with α = 0 correspond to homoclinic orbits in phase space. Forthe two cases which follow a non-trivial branch (α = 1 and 2) the flow approaches theunstable saddle node (qualitatively sketched in the figure) before departing from italong its unstable manifold. The trajectory in figure 6(b) then starts circling around thepoint which characterizes the fully developed turbulent state, with orbits of increasingsize. Before the end of the fifth orbit, it escapes through the unstable manifold of thesaddle node towards the laminar fixed point. Each orbit has an ellipsoidal shape andis made up of two portions with very small local radii of curvature, around whichthe flow spends most of its time, and two long portions of large radii of curvaturewhich are crossed very rapidly by the flow. Relaminarization is consistent with theemerging picture of shear flow turbulence as a transient event with a characteristiclifetime increasing exponentially with Reynolds number (Hof et al. 2006).

5. Discussion and conclusionsAlthough non-normality and transient growth are important issues, the traditional

emphasis on optimal disturbances may be misleading when trying to predict the onsetof transition. The results found here suggest that optimal initial perturbations in theform of steady streamwise vortices play a mostly passive role, while rapidly growingtravelling wave packets have the potential to induce transition past a reasonably lowthreshold value of the initial disturbance energy. This suggests that an energy-likefunctional is possibly not the most pertinent objective of the optimization procedure.

In the simulations performed here we have found that key to transition is theestablishment of a disturbed mean flow profile (mode α = 0), susceptible to anexponential or algebraic instability which causes enhanced growth of the travellingwave mode with α = ± 1. Such an early stage can be described by a weakly nonlineartriadic interaction model.

Once turbulence is established, its sustainment depends on non-normality, capableof producing rapid transient growth of the disturbance energy, and nonlinearity,which enables directional redistribution of the amplified disturbances. The couplingbetween transient growth and directional redistribution of energy causes the flowto orbit around the turbulent fixed point in the phase-space of figure 6. Near theedge of chaos the lifetime of turbulence is finite, and beyond a given value of t (afunction of the Reynolds number and of the streamwise dimension of the periodicdomain) relaminarization occurs. Interestingly, the edge state resembles the optimaldisturbance; this has recently been observed in pipe flow (Eckhardt et al. 2007; Pringle& Kerswell 2007) and still awaits an explanation.

The financial support of the EU (program Marie Curie EST FLUBIO 20228-2006)and of the Italian Ministry of University and Research (PRIN 2005-092015-002) aregratefully acknowledged.

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